Unitary element

In mathematics, an element of a *-algebra is called unitary if it is invertible and its inverse element is the same as its adjoint element.

Definition

Let ${\mathcal {A}}$ be a *-Algebra with unit $e$. An element $a\in {\mathcal {A}}$ is called unitary if $aa^{*}=a^{*}a=e$. In other words if $a$ is invertible and $a^{-1}=a^{*}$ holds.

The set of unitary elements is denoted by ${\mathcal {A}}_{U}$ or $U({\mathcal {A}})$.

A special case from particular importance is the case where ${\mathcal {A}}$ is a complete normed *-algebra, that satisfies the C*-identity ($\left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}$), which is called a C*-algebra.

Criteria

• Let ${\mathcal {A}}$ be a unital C*-algebra and $a\in {\mathcal {A}}_{N}$ a normal element. Exactly then is $a$ unitary if the spectrum $\sigma (a)$ consists only of elements of the circle group $\mathbb {T}$, i.e. $\sigma (a)\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid |\lambda |=1\}$.

Examples

• The unit $e$ is unitary.

Let ${\mathcal {A}}$ be a unital C*-algebra, then:

• Every projection, i.e. every element $a\in {\mathcal {A}}$ with $a=a^{*}=a^{2}$, is unitary. For the spectrum of a projection consists of at most $0$ and $1$, as follows from the continuous functional calculus.
• If $a\in {\mathcal {A}}_{N}$ is a normal element of a C*-algebra ${\mathcal {A}}$, then for every continuous function $f$ on the spectrum $\sigma (a)$ the continuous functional calculus defines an unitary element $f(a)$, if $f(\sigma (a))\subseteq \mathbb {T}$.

Properties

Let ${\mathcal {A}}$ be a unital *-algebra and $a,b\in {\mathcal {A}}_{U}$. Then:

• The element $ab$ is unitary, since ${\textstyle ((ab)^{*})^{-1}=(b^{*}a^{*})^{-1}=(a^{*})^{-1}(b^{*})^{-1}=ab}$. In particular, ${\mathcal {A}}_{U}$ forms a multiplicative group.
• The element $a$ is normal.
• The adjoint element $a^{*}$ is also unitary, since $a=(a^{*})^{*}$ holds for the involution *.
• If ${\mathcal {A}}$ is a C*-algebra, $a$ has norm 1, i.e. $\left\|a\right\|=1$.

See also

• Unitary matrix
• Unitary operator

References

• Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. pp. 57, 63. ISBN 3-540-28486-9.
• Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.
• Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3.